What is the pumping lemma for regular languages? 

The Pumping Lemma for Regular Languages provides a property that all regular languages must have, specifically that sufficiently long strings in the language can be "pumped" (repeated in a specific way) while remaining in the language. It is used to prove that certain languages are not regular by showing that they do not satisfy this property.

Application

To use the pumping lemma to prove that a language $L$ is not regular, you typically follow these steps:

1. Assume $L$ is regular and that the pumping lemma applies.
2. Choose a specific string $s$ in $L$ with $|s| \geq p$, where $p$ is the pumping length guaranteed by the lemma.
3. Show that no matter how $s$ is divided into $xyz$ satisfying the lemma's conditions, there exists some $i$ such that $xy^i z$ is not in $L$.
4. Conclude that $L$ cannot be regular because it contradicts the pumping lemma.